= Differential entropy under affine rescaling
{title2=$h_e(aX+b)=h_e(X)+\ln|a|$}
For real $a\ne0$ and $b$, the <change of variables formula> gives $f_{aX+b}(ax+b)=f_X(x)/|a|$. Therefore
$$
\boxed{h_e(aX+b)=h_e(X)+\ln|a|.}
$$
The absolute value also covers a decreasing transformation. If a numerical length changes from metres to centimetres, $a=100$ and the <differential entropy> changes by $\ln100\approx4.60517$ <nats>, although the physical uncertainty is the same. The reference coordinate scale must therefore accompany an absolute <differential entropy> value.
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